For the symmetric functions a_k and the definition of the triangles S_j(n,k) see a comment in A196841. Here x(j) = j for j = 1, 2, 3, 4 and x(j) = j + 1 for j = 5, ..., n. This is the triangle S_5(n,k), n >= 0, k = 0..n. The first five rows coincide with those of triangle A094638.